Recently there has been speculation concerning a link between very slowly decaying waves and non-linear instabilities in GR. We present a class of spacetimes which possess a certain geometric property (“stable trapping”), on which this very slow decay occurs. An example of this kind of spacetime is given by “ultracompact neutron stars”, and our results suggest that all stars with r < 3M might be unstable. Finally, we give an outline of the mathematical techniques used to prove that this slow decay occurs.